EPFL2016 Tacoma Narrows Bridge Torsion flutter · BLUFF BRIDGE DECKS 12 EPFL2016 Comparison of...

6/29/2016TACOMA NARROWS BRIDGE - 1 DOF TORSION FLUTTER

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EPFL2016

Tacoma Narrows Bridge – Torsion flutter

Allan Larsen


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Tacoma Narrows BridgeEPFL2016


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Assembly of bridge girder


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Tacoma Narrows Bridge – wind response and collapseEPFL2016

20 MAY 2015TACOMA NARROWS BRIDGE - STILL RESONATING AFTER 75 YEARS

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Areal photo 8 November 1940.EPFL2016

› March 1941 conclusion on aerodynamics of the committee:

› 1. As to torsion instability, it has been suggested that static moment of the wind on the floor of a bridge when tilted about its axis acts against the restoring moment of the cables, and at sufficiently high wind velocity can cause an overturning of the floor (static divergence). The wind tunnel tests indicates that this effect did not exist in the Tacoma Narrows Bridge.

› 2. However, convincing evidence has been found by oscillatory tests that beyond a certain wind velocity negative aerodynamic damping is to be expected.

› 3. There is no evidence for the formation of alternating vortices at a cross section similar to that used in the Tacoma Bridge, at least as long as the structure is not oscillating. It seems that it is more correct to say that the vortex formation and frequency is determined by the oscillation of the structure than that the oscillatory motion is induced by the vortex formation.

December 1940: Formation of the Carmody committee under FWA to investigate the couse of the collapse


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An unsolved flutter problem – 1 DOF torsion flutter

T. Wyatt, JWEIA 1992: "Most disappointing perhaps, is the continued lack of a deeper predictivemodel of the strong torsion phenomenon, extending beyond the direct empiricism of testing of a scale model"

Amman, Kármán, Woodruff: The failureof the Tacoma Narrows Bridge. A reportto the Honorable J. H. Carmody, 1941.

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BRIDGE DECK FLUTTER ANALYSIS7

2DOF flutter of modern bridge deck sectionsEPFL2016

0 0.1 0.2 0.3 0.41

10

100

62 m/s

0 m/s

15 m/s

35 m/s

50 m/s

Frequency [Hz]

Res

po

nse

R/R

st

Water tunnel tests to confirm vortex formation


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› Observation: A travelling vortex will form each time the cross section changes twist angle away from horizontal.

DVM simulation of the indicial function of the shallow H-section

29-06-2016A GENERIC MODEL FOR THE A2* INSTABILITY OF BLUFF BRIDGE DECKS

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EPFL2016

Larsen & Walther: JWEIA, 1997

10 11 12 13 14 15 160.4-

0.2-

0

0.2

0.4

Discrete vortex

0

Cm model

Non-dim. time tU/B

Cm

𝑇∗

𝐶𝑀(𝑡)

"Hand turned" discrete vortex analysis


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𝑈

𝑑Γ ≈ ½𝑈2𝑑𝑡Γ0

−Γ0

Γ0 =𝜋

4ℎ𝑈

Γ

𝐴= 𝑐𝑜𝑛𝑠𝑡.⟹

Γ0 + 𝑑Γ𝜋4ℎ2 + ℎ𝑑𝑥

=Γ0𝜋4ℎ2

⟹ 𝑑Γ = 𝑈𝑑𝑥

𝑑Γ ≈ ½𝑈2𝑑𝑡 ⟹𝑑𝑥

𝑑𝑡≈𝑈

2

𝑈𝑣 =1

2

𝑑𝑥

𝑑𝑡≈𝑈

4𝑇∗ ≈

𝐵

2

4

𝑈≈ 2

𝐵

𝑈𝐶𝑀 𝑡 =

1

21 −

𝑡

𝑇𝑡𝑈

𝐵

𝑀𝑣 𝑡 = 𝐹𝑣 𝑡𝐵

21 −

𝑡

𝑇

= ½𝜌𝑈3𝐵

21 −

𝑡

𝑇𝑡

𝐹𝑣 𝑡 = 𝜌𝑈Γ = 𝜌𝑈½𝑈2𝑡

𝑈𝑣

Work balance for sinusoidal torsion oscillation


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𝑊𝑣= 0

2∙𝑇

𝑀𝑣 𝑡 𝛼 𝑡 𝑑𝑡 = 0

2∙𝑇

𝑀𝑣 𝑡 𝜔𝛼cos(𝜔𝑡)𝑑𝑡 =

𝜌𝐵𝑈3𝛼𝜔

4 0

𝑇

1 −𝑡

𝑇𝑡 cos 𝜔𝑡 𝑑𝑡 −

1

2 0

𝑇

1 −𝑡 + 𝑇

𝑇(𝑡 + 𝑇) cos 𝜔(𝑡 + 𝑇 ) 𝑑𝑡

Work exerted by vortex:

𝑊𝑑 =𝛿

𝜋𝐼𝜔3𝛼2

0

2∙𝑇

cos2 𝜔𝑡 𝑑𝑡Work exerted by viscous damping force:

The equivalent viscous damping of the vortex: 𝑊𝑣 = 𝑊𝑑

𝛿𝑎 =𝜋

2𝛼

𝜌𝐵4

𝐼

𝑈∗

2𝜋

3𝑈∗

4𝜋sin

4𝜋𝑈∗ 3 − 2cos

4𝜋𝑈∗ +

12

cos8𝜋𝑈∗ − cos

4𝜋𝑈∗ − 1

8𝜋𝑈∗ +

12sin

16𝜋𝑈∗

Where: 𝑈∗ =𝑈

𝑓𝛼𝐵


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Comparison of analytical model to wind tunnel data

Amman, Kármán, Woodruff: The failureof the Tacoma Narrows Bridge. A reportto the Honorable J. H. Carmody, 1941.

Tacoma Narrows: 𝜇 = 𝜌 𝐵2 =0.0207

Present

Larsen, Larose: JSV 334, 2015

𝑈𝑐𝑟 = 𝑓𝛼𝐵 4 +𝜋3𝛼

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𝐼

𝜌𝐵4𝛿𝑠

Slope at 𝑈∗ = 4

𝛼 = 0.175 𝑟𝑎𝑑 10 𝑑𝑒𝑔 , 𝛿𝑠=2𝜋𝜁𝑠

𝑈𝑐𝑟 = 𝑓𝛼𝐵 4 + 11.5𝐼

𝜌𝐵4𝜁𝑠

1DOF equivalent of Selberg's formula

DVMFLOW simulation of free response of elastically suspended Tacoma section at U/fB=2 and U/fB=6


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A blast from the past – Tacoma Narrows footage


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A generic model for the 𝐴2∗ instability of

bluff bridge decks Scanlan & Tomko: ASCE EM6, 1971


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𝐴2∗ aerodynamic derivative – DVM simulations

𝐴2∗ 𝑈∗ =

1

32𝜋3𝑈∗ 3

𝛼

𝑈∗

4𝜋sin

4𝜋𝑈∗ 3 − 2cos

4𝜋𝑈∗ +

12

cos8𝜋𝑈∗ − cos

4𝜋𝑈∗ − 1

8𝜋𝑈∗ +

12sin

16𝜋𝑈∗

𝐴2∗ = ( 𝛿𝑎 𝜋)( 𝐼 )𝜌𝐵4𝐴2

∗ is non-dimensional aerodynamc damping:

10 deg.

20 deg.

45 deg.


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In the wake of Galloping Gertie


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The University of Washington 1:50 wind tunnel tests


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The University of Washington 1:50 wind tunnel tests


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Stills from the vortex camera wind speed: 3 ft/s

› UW Bullitin No. 116, Part III, p. 121: Close scrutiny of th upwind sidewalk areawill reveal a portion of the upward flow through the open deck entrapped by the clockwise rotation of the vortex originating from the top of the girder. It is not difficult to imagine a considerable alteration in the pressure on thissidewalk as a result of the flow revealed at this instant.


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Flow visualization, wind speed approximately 3 ft/s


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Smoke tests at the bridge site


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The engineering solution to the torsion instability: Add 11 m deep truss, remove 2.4 m edge beams

Cost: $ 6.6 millionWeight suspension cables: 3817 tWeight carried by cables: 11250 t

Cost: $ 14.0 millionWeight suspension cables: 5441 tWeight carried by cables: 18160 t


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EPFL2016

Tacoma Narrows Bridge - vortex excitation mitigation

EPFL2016 Tacoma Narrows Bridge Torsion flutter · BLUFF BRIDGE DECKS 12 EPFL2016 Comparison of...

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